Help needed for deciding on Omega and Zeta skills and general feedback on direction

If you are planning on building an empire team, I'd definitely zeta Palpatine's leadership first. With Vader's leadership you can solo the pit raid, but Palpatine gives you more edge in arena, TW, TB and so on.
Thrawn is a must have for almost any top tier team composition (especially an empire team) and his unique is a solid zeta, but it's not a game changer and I wouldn't consider it a top 10 but even top 20 zeta. For now I'd advise to go with Palpatine.

I would also recommend to focus on one team at once while having the side projects in mind. If you decided to go with empire, gear up a team 5 first (Vader, Palpatine, Thrawn is the core and I'd go with Tarkin and Tie Pilot to develop in fleet arena as well). Meanwhile, you should farm the characters for CLS hero's journey (Leia, Old Ben, ST Han, Farmboy Luke and 5 empire toons to unlock R2). BHs will help you get Chewy (solid late game arena team with Raid Han and CLS), Old Republic is needed for Revan and you'll want to add JTR to your roster at some point.

@RandomSithLord Thank you for the response!

Based on what you said, it sounds like I'm on the right path with most of what I'm doing; my Empire team at the moment is EP, Vader, Tarkin, TFP and Dooku as my temporary 5th, up until I get Thrawn.

I'm also farming Old Ben and working on STHan in preparation for CLS based on some in-game recommendations, and I already have a side team of BH's, so it seems like I'm almost ready for R2 and Chewie and hopefully CLS before he comes around again!

I'm curious, is it better to pick out a Zeta first before going crazy with Omega's? Palpatine's Zeta seems amazing, particularly with the amount of debuffs the whole team toss out, but some of the Omega's seem awesome too, like Palpatine's Power of the Darkside and Vader's Force Crush..

They're both important. Your arena team will need all omega's to be truly good.
At the stage you're in zeta's will become a farfetched something. It'll take 2 months to gather 1, so don't think about them too much ftm.
Although, palps lead will be a gamechanger for your team, so you'll really need that one.

Do not give zeta on Vader’s lead.
It’s not worthy for you right now.

Palpatine Lead seems the best.
Few days later, you can get Thrawn.
Make arena squad with zEP DV Thrawn GMT TFP. Those are empire characters which you have.
Arena is the most important thing for you because arena is the main resource of crystal.

@Moorebid Thank you for the response :) I could get Palpatine's in about a week (just need 5 or so more omega shards), so it sounds like that's what I should go with next. It's going to be hard saving them up when there are so many omega skills I really want!

@komji15 Thanks again! I really can't wait for Thrawn tomorrow night.The last 3 months or so of character farming I've done has been preparing for Palpatine and Thrawn; it's great to see it paying off. I'm glad I'm starting to get a competitive team together too - my current BH team that I run in the arena is really showing it's lack of competitiveness without Bossk.

Once I pick up EP's zeta, is there an order of importance for the next skills I should get for the characters in that squad (of EP, Vader, Thrawn, GMT, TFP)? Or should I just go with what I think will be most helpful?

Thrawn’s unique is worthy. Tarkin’s... is good but not your priority. You should farm Nihilus and Sion. Sion’s unique zeta is must-have.

So, I never actually went with Palp's lead zeta, but my shard (June 18) was dominated by them until zBastila teams took over. You'll get a lot of mileage from it, especially since your shardmates wouldn't have gotten her as a marquee.

Have you unlocked the T3 Ship Ability Mats Challenge? If not, that should be your fleet currency priority -- building your fleet up to the point you can at least 3* that so you have a shot at zetas from it. Without it, zeta abilities will still be a ways away for you. I sank currency into zeta mats directly before having it on farm and that set me back a fair amount.

Regarding omegas, they are absolutely still very important, although usually less impactful than zetas. For you, I would zeta EP's lead once you're able to. In the mean time, I would run EP (L), Vader, Tarkin, TFP, and Thrawn once he's unlocked. I would omega Thrawn's specials first probably since each has its cooldown reduced by one, which is huge. Then I'd do all of Vader's, they're all really good. Then go back and do Thrawn's basic. Then EP's basic and first special, TFP's TIE Strike, and then if you want to stick with this roster I'd do Tarkin's omegas. They're good, with Special 1 being the best in most situations. Special 2 is only rather important if you use him on a team without EP lead. Basic is good for assault battles and TBs and the like since the stacking potency carries between rounds, but in arena and TW you won't notice a whole lot of difference.

E: regarding further zetas, I agree with @komji15. Farm Nihilus and Sion once you unlock their nodes, and once they're geared up well and around 4* you should swap them for TFP and GMT. Sion's zeta is a must then. Then Thrawn's Ebb & Flow, then go on to another team as none of the other zetas are useful enough to do unless can afford a luxury zeta.

@komji15 I'm currently working on farming Sion (very slowly), and will also farm Nihilus when I hit 85 and can unlock his node - that was my plan, at least!

@Jbnf That's funny, I was just preparing to get 1 more Dark Side ship up to 4* so that I can get Tarkin's ship to 5*, so I can start doing that event (whew, that was tedious to type!) and I'm having a real hard time trying to decide which would be the easiest / best pick for a 5th ship required. Currently I have Slave 1 at 5*, Vader's TIE Advanced 4*, KRU's Silencer 4*, ITF 5* and Genosian Soldier's Ship at 2* just in my Dark Side fleet, so naturally I'd probably try to get the Genosian up, but I was also thinking about Maul's Scimitar or Cad's Xanadu Blood.. The Xanadu would be the most tedious and would interfere with my Sion farming, though, so that seems like a bad idea.

As for the omega's, thanks for all the advice! I went crazy when I got a large chunk of omega shards from the Thanksgiving handout (I hope we get more for Christmas!) and picked up 2 of Tarkin's Omega skills (his basic and his 1st special); I'll wait while you and others yell at me for being overeager.

I definitely planned to get Sion (am already farming him) and Nihilus (when I hit 85) and to start a Sexy Sith Squad, but that's probably a couple months off.. Definitely something for the future though!

I agree with pretty much everyone's suggestions. I would also add, plan zetas/omegas for up coming events. eg, bb8, cls, thrawn, etc. If the bb8 event is near and yah don't have a great FO team ready, have that zeta ready for kylo because he can essentially solo the event.
https://swgoh.gg/p/667674687/zetas/
Those are my zetas. The only one somewhat questionable is visas, but got a lot of use out of it.